Let F be the vector field F=(13x^2 y+3y^3 -y)i -12x^3 j. Find the maximum value of ∫F•dr where c is positively oriented simple closed curve.

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Let F be the vector field F=(13x^2 y+3y^3 -y)i -12x^3 j. Find the maximum value of ∫F•dr where c is positively oriented simple closed curve.

Mathematics
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You may start with green's thm I think
\[\oint F.dr = \iint\limits_{R} \text{curl}(F)~dA = \iint\limits_{R} -49x^2-9y^2+1~dA \]

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Easy to see that the integrand stays positive in the region \(49x^2+9y^2 \lt 1\), so should we pick this as region of integration ?
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Then we're correct, go ahead and evaluate the integral over that ellipse
\[max(\oint F.dr) ~= ~\iint\limits_{49x^2+9y^2\lt 1}~~1-49x^2-9y^2~dA\]
you will need to use change of variables
I'm getting \(\dfrac{21\pi}{2}\) http://www.wolframalpha.com/input/?i=21*%5Cint_0%5E%282pi%29%5Cint_0%5E1+%281-r%5E2%29*r+dr+d%5Ctheta
looks correct to me, I got \[\frac{ 42\pi }{ 4 }\]

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